Transcript The new electronic ballast generation

Passivity based control applied to power converters
Passivity based control applied to power
converters
Marco Liserre
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Marco Liserre
[email protected]
Passivity based control applied to power converters
Passivity based control history


70’s definition of dissipative systems (Willems)
1981 application to rigid robots (Arimoto e Takagi)
in power electronics . . .
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1991 first theoretical paper (Ortega, Espinoza & others)
1996 first experimental paper (Cecati, & others, IAS Annual Meeting)
1998 first book “Passivity-Based Control of Euler-Lagrange Systems”
(Ortega and Sira-Ramirez, Springer, ISBN 1852330163)
1999 application to active filters (Mattavelli and Stankovic, ISCAS 99)
2002 Brayton-Moser formulation (Jeltsema and Scherpen, Am. Control
Conf.)
2002 application to multilevel converters (Cecati, Dell’Aquila, Liserre,
Monopoli, IECON 2002)
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Passivity based control applied to power converters
Contribution of my research group on the topic

collaborations with:
 University of L’Aquila (Prof. Cecati)
 University of Delft (Prof. Scherpen)

main papers:
 C. Cecati, A. Dell’Aquila, M. Liserre, V. G. Monopoli “A passivity-based
multilevel active rectifier with adaptive compensation for traction
applications” IEEE Transactions on Industry Applications, Sep./Oct. 2003,
vol. 39, no. 5.
 A. Dell’Aquila, M. Liserre, V. G. Monopoli, P. Rotondo “An EnergyBased Control for an n-H-Bridges Multilevel Active Rectifier” IEEE
Transactions on Industrial Electronics, June 2005, vol. 52, no. 3.
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Passivity based control applied to power converters
Basic idea of the Passivity-based approach

The basic idea of the PBC is to use the energy to describe the state of the
system

Since the main goal of any controller is to feed a dynamic system through
a desired evolution as well as to guarantee its steady state behavior, an
energy-based controller shapes the energy of the system and its
variations according to the desired state trajectory

If the controller is designed aiming at obtaining the minimum energy
transformation, optimum control is achieved

The PBC offers a method to design controllers that make the system
Lyapunov-stable

The “energy approach” is particularly suitable when dealing with:
 electromechanical systems as electrical machines
 grid connected converters (non-linear model)
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Passivity based control applied to power converters
The introduction of damping

The control objective is usually achieved through an energy reshaping
process and by injecting damping to modify the dissipation structure of
the system

From a circuit theoretic perspective, a PBC forces the closed-loop
dynamics to behave as if there are artificial resistors — the control
parameters — connected in series or in parallel to the real circuit
elements

When the PBC is applied to grid connected converters, harmonic
rejection is one of the main task, hence the passive damping can be
substituted by a dynamic damping (i.e. virtual inductive and capacitive
elements should be added)

The point of view is always the energy reshaping (i.e. the energy
associated to the harmonics)
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Passivity based control applied to power converters
The Eulero-Lagrange formulation

Passivity-based control has been firstly developed on the basis of EuleroLagrange formulation

One of the major advantages of using the EL approach is that the
physical structure (e.g., energy, dissipation, and interconnection),
including the nonlinear phenomena and features, is explicitly
incorporated in the model, and thus in the corresponding PBC

This in contrast to conventional techniques that are mainly based on
linearized dynamics and corresponding proportional-integral–derivative
(PID) or lead–lag control
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Passivity based control applied to power converters
The Passivity Based Controller design

In the context of EL-based PBC designs for power converters, two
fundamental questions arise:
 which variables have to be stabilized to a certain value in order to
regulate the output(s) of interest toward a desired equilibrium
value? In other words, are the zero-dynamics of the output(s) to be
controlled stable with respect to the available control input(s), and
if not, for which state variables are they stable?
 where to inject the damping and how to tune the various
parameters associated to the energy modification and to the
damping assignment stage?
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Passivity based control applied to power converters
Dissipativity definition
dissipativity
definition
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Passivity based control applied to power converters
Passivity definition
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Passivity based control applied to power converters
Definitions
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Supply Rate: speed of the energy flow from a source to the system
Storage function: energy accumulated in a system
Dissipative systems: systems verifying dissipation inequality:
“Along time trajectories of dissipative systems the following relationship holds:
energy flow ≥ storage function”
(In other words, dissipative systems can accumulate less energy than that supplied by
external sources)

The basic idea of PBC is to shape the energy of the system according to a desired
state trajectory, leaving uncontrolled those parts of the system not involved in
energy transformations, this result can be obtained only working on “strictly
passive” systems
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Passivity based control applied to power converters
Feedback systems decomposition

dividing the system into simpler subsystems, each one identifying those parts of
the system actively involved in energy transformations

each subsystem has to be passive introducing energy balances, expressed in
terms of the Eulero-Lagrange equations
passivity invariance
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Passivity based control applied to power converters
Feedback systems decomposition

The full order model describing the system is divided into simpler subsystems
identifying those parts actively involved in energy transformation

Hence, energy balances, expressed in terms of the Eulero-Lagrange equations
(based on the variational method and energy functions expressed in terms of
generalised coordinates), are introduced

The system goes in the direction where the integral of the Lagrangian is
minimized (Hamilton's principle)
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Passivity based control applied to power converters
Feedback systems decomposition

This formulation highlights active, dissipative and workless forces i.e. the active
parts of the system (those which energy can be modified by external forces),
those passive (i.e. dissipating energy, e.g. thermal energy), and those parts
which do not contribute in any form to control actions and can be neglected
during controller design

Because of the energy approach, it is quite straightforward to obtain fast
response under condition that the control "moves" the minimum amount of
energy inside the system

Moreover, because global stability is ensured by passivity properties, a simple a
effective controller can be designed
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Passivity based control applied to power converters
Eulero-Lagrange formulation

The eulero-lagrange formulation is particularly suited for the control of
electromechanical systems as electrical motors

In fact different subsystems are related by their ability to transform energy,
therefore it is a good thing to define energy functions for each one, expressed in
terms of generalised coordinates qi.

In electric motor case:
qm mechanical position (for mechanical subsystems)
qe electric charge (for electrical subsystems)

Using variational approch we can introduce Lagrangian equations of the system
and apply Hamilton's principle. This method highlights subsystems
interconnections and their various energies: dissipated, stored and supplied
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Passivity based control applied to power converters
Eulero-Lagrange formulation
induction motor formulation
The mechanical subsystem does not take an active part in control actions, i.e.
it doesn't produce energy but only transforms and dissipates the input energy,
for design purposes its contribution can be considered as an external
disturbance for the electrical subsystem and the controller has to compensate
for this disturbance, in order to maintain electrical equation balance. In
“passivity terms”, it defines a passive mapping around the electrical
subsystem, it can be neglected during controller design and the attention can
be focused on the electrical subsystem.
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Passivity based control applied to power converters
Eulero-Lagrange formulation
The electrical subsystem is simply passive, then its evolution can be corrupted by
any external disturbance leading to instability. Therefore, in order to obtain global
stability, it is an important step of the approach to make it strictly passive by
means of the addition of a suitable dissipative term (damping injection)
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Passivity based control applied to power converters
Passivity-based control of the H-bridge converter
 PBC has been successfully applied to d.c./d.c. converters, active rectifiers
and multilevel topologies
 Particularly the single-phase Voltage Source Converter (VSC) also called
H-bridge or full bridge can be used as universal converter due to the
possibility to perform dc/dc, dc/ac or ac/dc conversion
 Moreover it can be used as basic cell of the cascade multilevel converters
 In the following it will be reviewed the application of the PBC to H-bridge
single phase inverters (one-stage and multilevel) using the Brayton-Moser
formulation which is the most suitable for the converter control
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Passivity based control applied to power converters
Passivity-based control of the H-bridge converter

Control of one H-bridge-based active rectifier
G. Escobar, D. Chevreau, R. Ortega, E. Mendes, “An adaptive passivity-based
controller for a unity power factor rectifier”, IEEE Trans. on Cont. Syst. Techn., vol.
9, no. 4, July 2001, pp. 637 –644
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Control of two (multilevel) H-bridge-based active rectifier
C. Cecati, A. Dell'Aquila, M. Liserre and V. G. Monopoli, "A passivity-based
multilevel active rectifier with adaptive compensation for traction applications",
IEEE Trans. on Ind. Applicat., vol. 39, Sept./Oct. 2003 pp. 1404-1413
the two dc-links are not independent !
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Control of n (multilevel) H-bridge-based active rectifier
A. Dell’Aquila, M. Liserre, V. G. Monopoli, P. Rotondo “An Energy-Based Control
for an n-H-Bridges Multilevel Active Rectifier” IEEE Transactions on Industrial
Electronics, June 2005, vol. 52, no. 3.
the n dc-links are independent !
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Passivity based control applied to power converters
Brayton-Moser Equations

Brayton and Moser, introduced in 1964 a scalar function of the voltages across
capacitors and the currents through inductors in order to characterize a given
network
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This function was called the Mixed-Potential Function P(iL, vC) and it allows to
analyze the dynamics and the stability of a broad class of RLC networks
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These equations can be considered an effective alternative to Euler-Lagrange
formulation
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This formulation has a main advantage over the counterpart in case of power
converter control: it allows the controllers to be implemented using measurable
quantities such as voltages and currents.
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Passivity based control applied to power converters
Brayton-Moser Equations
Topologically Complete Networks = networks which state variables form a complete
set of variables
Complete Set of Variables = set of variables that can be chosen independently without
violating Kirchhoff’s laws and determining either the current or voltage (or both) in
every branch of the network
Additionally for Topologically Complete Networks it is possible to define two
subnetworks
One subnetwork has to contain all inductors and current-controlled resistors
The other has to contain all capacitors and voltage controlled resistors
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Passivity based control applied to power converters
Brayton-Moser Equations
For the class of topologically complete networks it is possible to construct the mixedpotential function directly. For this class it is known that the mixed potential is of the
form:
P( iL ,vC )  R( iL )  G( vC )  N( iL ,vC )
R(iL) is the Current Potential (Content) and is related with the current-controlled
resistors and voltage sources
G(vC) is the Voltage Potential (Co-content) and is related with the voltage-controlled
resistors and current sources
N(iL,vC) is related to the internal power circulating across the dynamic elements
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Passivity based control applied to power converters
Brayton-Moser Equations
The components of the Mixed-Potential Function can be analysed in more detail as
follows:
P( iL ,vC )   PR ( iL )  PE ( iL )   PG ( vC )  PJ ( vC )   PT ( iL ,vC )
R
G
N
PR is the Dissipative Current Potential
PG is the Dissipative Voltage Potential
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Passivity based control applied to power converters
Brayton-Moser Equations
The dissipative current and voltage potentials can be calculated as follows:
iL
PR ( iL )   vR ( i' L )di' L
0
vC
PG ( vC )   iG ( v'C )dv'C
0
In case of linear resistor PR is half the dissipated power expressed in terms of inductor
current, and PG is half the dissipated power expressed in terms of capacitor voltages.
P( iL ,vC )   PR ( iL )  PE ( iL )   PG ( vC )  PJ ( vC )   PT ( iL ,vC )
R
G
N
PE is the total supplied power to the voltage sources E
PJ is the total supplied power to the current sources J
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Passivity based control applied to power converters
Brayton-Moser Equations
P( iL ,vC )   PR ( iL )  PE ( iL )   PG ( vC )  PJ ( vC )   PT ( iL ,vC )
R
G
N
PT is the internal power circulating across the dynamic elements and is represented by:
PT ( x )  i  vC
T
L
In this representation  denotes the interconnection matrix and it is determined by KVL
and KCL
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Passivity based control applied to power converters
Brayton-Moser Equations
Finally the expression of the mixed-potential function
P( iL ,vC )   PR ( iL )  PE ( iL )   PG ( vC )  PJ ( vC )   PT ( iL ,vC )
R
G
N
can be rewritten as follows:
P( x )  PD ( x )  PT ( x )  PF ( x )
PD(x)= PR(x)- PG(x) is the Dissipative Potential
PF(x)= PJ(x)- PE(x) is the Total Supplied Power
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Passivity based control applied to power converters
Brayton-Moser Equations
The dynamic behaviour of topologically complete networks is governed by the following
differential equations :
diL P( iL ,vC )
 L( iL )

dt
iL
dvC P( iL ,vC )
C( vC )

dt
vC
iL = (iL1 , . . . , iL )T are the currents through the  inductors
vC = (vC1 , . . . , vC )T are the voltages across the  capacitors.
These differential equations correspond with Kirchhoff’s voltage and current laws
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Passivity based control applied to power converters
Brayton-Moser Equations
The previous equations can be expressed in a more compact way as follows:
 P( x ) P( x ) 
 P( x ) 
Q( x )x   x P( x )  

...



x

x

x


1
n 

T
with the state vector xRn = R+ defined as
T
 iL 
x 
 vC 
and with the diagonal square matrix Q(x)  R(+)x(+) defined as
0 
  L(iL )
Q( x)  

C (vC ) 
 0
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Passivity based control applied to power converters
Brayton-Moser Equations
When a circuit contains only linear passive inductors and capacitors, then the diagonal
matrices L(iL)  Rx and C(vC)  Rx are of the form:
 L1
0
L( iL )  
.

 0
0
0
L2
.
0
.
0
0
0
0 
.

L 
C1 0
0 C
2
C( vC )  
 .
.

 0 0
0
0 0 
. . 

0 C 
0
The Brayton-Moser equations are closely related to the co-Hamiltonian H*(iL, vC) (that
represents the total co-energy stored in the network).
If the co-Hamiltonian is known, then the matrices L(iL) and C(vC) can be easily found
as follows
L(iL )  i2L H * (iL , vC ), C(vC )  v2C H * (iL , vC )
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Passivity based control applied to power converters
Switched Brayton-Moser Equations
For a circuit with one or more switches it is possible to obtain a single Switched
Mixed-Potential Function by properly combining the individual mixed-potential
functions associated to each operating mode.
u=0
P0(x)
u=1
P1(x)
Then it is possible to obtain one Switched Mixed-Potential Function parameterized by
u as
P(u, x)  (1  u)P0 ( x)  uP1 ( x), u 0, 1
The Switched Mixed-Potential Function is consistent with the individual MixedPotential Functions
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Passivity based control applied to power converters
Switched Brayton-Moser Equations
It is worth to notice that the only difference between each individual Mixed-Potential
Function and the Switched Mixed-Potential Function will be in the term
PT ( x )  iLT vC
and in particular in the interconnection matrix  which becomes a function of u,  (u)
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Passivity based control applied to power converters
Average State Model
When the switching frequency is sufficiently high, it is possible to prove that the
average state model of a circuit with a single switch can be derived from the discrete
model by only replacing the discrete variable u{0,1} with the continuously
varying duty-cycle variable μ[0,1] . Additionally, to show that the model is a state
average model, the state vector x is replaced by the state average vector z
μ
u
Average
Discrete
State
Model
z
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x
Model
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Passivity based control applied to power converters
Average State Model
The former result can be extended to circuits with multiple switches. In this case
the matrix  (u) assumes as many configurations as the possible combinations of the
status of the switches are (e. g. for an H-bridge converter  is a mono-dimensional
matrix and may assume three distinct values {-1,0,1})
μi
ui
Discrete
State
Model
zi
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Average
xi
Model
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Passivity based control applied to power converters
Passivity Based Control – Procedure
To design a Passivity Based controller the average co-energy function H*(z) and the
dissipative potential PD(z) have to be modified. To this purpose the BraytonMoser equations can be rewritten as:
PT ( z ) PD ( z ) PF ( z )
Qz 


z
z
z
f(z)
constant
The first two derivative terms are still
function of z, in the sense that the partial
derivatives of PT(z) and PD(z) are still
dependent on z
The third term is constant meaning that
the partial derivative of PF(z) is not
dependent on z anymore
The following step is to rewrite the previous equations by replacing the state variables z
with an auxiliary system of variables ξ which represent the desired state trajectories
for inductor currents and capacitor voltages:
PT (  ) PD (  ) PF (  )
Q 





f(ξ)
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constant
The first two derivative terms are still
function of ξ
The third term is constant and is
obviously equal to the partial derivative
of PF(z)
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Passivity Based Control – Procedure
being z
= z − ξ the average state errors, it is possible to write:
Qz  Qz  Q 
PT ( z )
P ( z )
P (  )
P (  )
( z ) D
( z ) T
( ) D
( )
z
z


Assuming that the first two derivatives are linear functions of z and the last two
derivatives are linear functions of ξ, yields:
Qz 
PT ( z   ) PD ( z   ) PT ( z ) PD ( z )



( z   )
( z   )
( z )
( z )
The previous expression represents the error dynamics and it could be obtained from
Qz 
PT ( z ) PD ( z ) PF ( z )


z
z
z
by simply replacing the variable z with the error variable z
derivative of PF
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and eliminating the
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Passivity based control applied to power converters
Passivity Based Control – Procedure
The next step is to add a damping term to the error dynamics to ensure asymptotic
stability
P ( z ) PD ( z ) PV ( z )
Qz  T


z
z
z
This injection can be seen as an
expansion of the dissipative potential
Considering z = (i L, v C)T where
i L = (z 1 . . . z )T are the error-currents through the inductors
v
C=
(z
+1
...z
+)
T
are the error-voltages across the capacitors
The injected dissipation can be decomposed as follows:
PV ( z )  PVr ( iL )  PVg ( vC )
The injected dissipation together with the dissipative potential of the system, gives the
Total Modified Dissipation Potential PM
PM ( z )  PD ( z ) z  z  PV ( z )
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Passivity based control applied to power converters
Passivity Based Control – Procedure
Subtracting
Qz 
PT ( z ) PD ( z ) PV ( z )


z
z
z
Qz 
PT ( z ) PD ( z ) PF ( z )


z
z
z
from
the controller dynamics are obtained
PF ( z ) PT (  ) PD (  ) PV ( z )
Q  



z


z
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Passivity based control applied to power converters
Passivity Based Control – Procedure
Two theorems ensure the stability of the closed loop system.
THEOREM 1 (R-Stability)
If RS is a positive-definite and constant matrix, and
1
2
1
L RS  (  )C
with 0 << 1, then for all (i
Qz 
tend to zero as t → ∞
L,

v
1
2
 1
C)
the solutions of
PT ( z ) PD ( z ) PV ( z )


z
z
z
where closed-loop resistance matrix RS is
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 2 PM ( z )
RS ( iL ) 
2
iL
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Passivity Based Control – Procedure
THEOREM 2 (G-Stability)
If GP is a positive-definite and constant matrix, and
1
2

1
C GP  T (  )L
with 0 << 1, then for all (i
Qz 
tend to zero as t → ∞.
L,
v
1
2
 1
C)
the solutions of
PT ( z ) PD ( z ) PV ( z )


z
z
z
where closed-loop conductance matrix GP is
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 2 PM ( z )
GP ( vC )  
2
vC
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Passivity Based Control – Procedure

With these theorems lower bounds are found for the control parameters (RS
and/or GP )

These lower bounds ensure a ”reasonably nice” response in terms of overshoot,
settling-time, etc

If just one of these theorems is satisfied, the system is stable. This means there
are two damping injection strategies that can be selected:
Series Damping
(damping on
inductor currents)
PM ( z )  PD ( z ) z z  PVr ( iL )
Parallel Damping
(damping on
capacitor voltages)
PM ( z )  PD ( z ) z  z  PVg ( vC )
Although it is sufficient to use only one of these strategies, the equations could
contain both the series damping injection term and the parallel damping injection
term
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Passivity based control applied to power converters
Passivity Based Control – Procedure
Finally, if n is the number of minimum phase states it is possible to modify n
equations of the + differential equations in
PF ( z ) PT (  ) PD (  ) PV ( z )
Q  



z


z
To this purpose n minimum phase states have to be found. Consequently the
remaining +-n state variables will be indirectly controlled through the control of
the n selected states
For the n selected variables it is possible to set the derivative of reference value to
zero obtaining n algebraic equations:

PF ( z ) PT (  ) PD (  ) PV ( z )



0
zi
i
i
zi
and +-n differential equations:
Qi  
Marco Liserre
PF ( z ) PT (  ) PD (  ) PV ( z )



zi
i
i
zi
Controller
Equations
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Passivity based control applied to power converters
Passivity Based Control – Procedure
Controller Implementation
At the beginning initial values of the n control inputs have to be set
Using these values the differential equations
PF ( z ) PT (  ) PD (  ) PV ( z )
Qi  



zi
i
i
zi
can be solved to obtain the time evolution of the auxiliary variables
for the indirectly controlled variables.
The former references are needed to solve the algebraic equations

PF ( z ) PT (  ) PD (  ) PV ( z )



0
zi
i
i
zi
which solutions are the set of values for the control inputs to be applied in
the next switching period.
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge
+DC
T1
L
T3
a
i
R
C
e
b
T2
T4
-DC
grid
The passivity-based controller will be designed by
inspection, identifying the potential functions
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge
L
PR  0
+DC
z1
T1
T3
2
2
z
PG 
2R
a
i
R
C
e
b
T2
z2
PD  PR  PG
T4
-DC
PJ  0
PE  ez1
PF  PJ  PE
z22
PD  
2R
PF  ez1
PT  z1    z2
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge
L
2
2
+DC
z1
T1
T3
a
i
R
C
e
b
T2
T4
-DC
z2
z
PD  
2R
PF  e  z1
PT  z1    z2
P
   z2  e
z1
P
z2
     z1
z2
R
Marco Liserre
2
2
z
P    z1    z2  e  z1
2R
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Passivity based control applied to power converters
PBC of an H-bridge
+DC
z1
L
T1
T3
a
i
R
C
e
b
T2
T4
z2
 Lz1    z2  e LKT
z2
Cz2      z1 LKC
R
-DC
 Lz1    z2  e
 L1     2  e
z2
Cz2      z1
R
C 2  
Marco Liserre
2
R
   1
controller
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge: damping injection
+DC
z1
L
T1
T3
a
i
R
C
e
b
T2
T4
-DC
 Lz1    z2  e
Cz2  
z2
   z1
R
 Lz1    z2  e  RDS z1
z2
z2
Cz2      z1 
R
RDP
Marco Liserre
z2
RDS z12
z22
PV ( z ) 

2
2RDP
z22 RDS z12
z22
PM ( z )  


2R
2
2RDP
R( z1 )  RDS
1
1
G( z2 )  
R RDP
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Passivity based control applied to power converters
PBC of an H-bridge: damping injection
 Lz1    z2  e
+DC
z1
T1
L
T3
a
i
R
C
e
b
T2
z2
z2
Cz2      z1
R
T4
-DC
 Lz1    z2  e  RDS z1
z2
z2
Cz2      z1 
R
RDP
 L1     2  e  RDS z1
2
z2
C 2      1 
R
RDP
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge: control variables
+DC
z1
L
T1
 L1     2  e  RDS z1
T3
a
i
R
C
e
b
T2
T4
z2
2
z2
C 2      1 
R
RDP
-DC
 which variables have to be stabilized to a certain value in order to regulate
the output(s) of interest toward a desired equilibrium value?
 in other words, are the zero-dynamics of the output(s) to be controlled
stable with respect to the available control input(s), and if not, for which state
variables are they stable?
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge: zero-dynamics
 The steady-state solution in case of direct control of the dc-voltage or in
case of indirect control of the dc-voltage (through the grid current) should
be found
Switching function in case of direct control
Switching function in case of indirect control
2
1.5
1.5
1
0.5
0.5
Funzione di switching s
Funzione di switching s
1
0
-0.5
0
-0.5
-1
-1
-1.5
-2
0
0.002
0.004
0.006
0.008
0.01
Tempo [s]
0.012
0.014
0.016
0.018
1 2
 eR  
     R

L
V
d 
 
0.02
-1.5
0

0.05
0.1
0.15
0.2
0.25
Tempo [s]
0.3
0.35
0.4
0.45
0.5
  2 Rz1  Lz1  RLCz1  e  RCe
CR
Lz1  e
 A stable system can be obtained only by indirectly controlling the dc
voltages through the ac current i*
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge
 A stable system can be obtained only by indirectly controlling the
dc voltages through the ac current i*
 This means that
as i  i*,
vc  ξ2  Vd
 From the power balance it results that
dc voltage reference
2Vd2 
Id 
E
load conductance
1

R
grid voltage amplitude
controller
and i*  I sin t
 
d
 Lz*1     2  e  RDS z1
C 2   2    z*1  GDP z2
reference
reference
voltage Vd power balance
current i*
and load
conductance θ
Marco Liserre
ODE
internally algebraic switching
function µ
generated ξ2
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Passivity based control applied to power converters
PBC of an H-bridge
 Since it is possible to control directly the grid current,
z1 
1
(e  Rz1 )   z2 
L
average KVL
on the a.c. side

1
2
 e  Rz  Lz
1
*
1
 RDS z1

predictive action
+
damping injection
 The d.c. voltage is controllable only indirectly, through an internal variable ξ2
z2 
1
  z1   z2 
C
average KCL
on the d.c. side
differential eq.
1
+
ˆ G z
2   z1*  
2
DP 2
C
damping injection

 Then it is necessary to estimate the d.c. load
Marco Liserre




ˆ   ˆ  c2 ˆ  c1  2 z2
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge: damping tuning
1
2
1
L RDS  (  )C

1
2
L 
 1
C RDS
 1
RDS 

1
L
C
L
+DC
T1
T3
RDS
a
i
R
C
e
GDP
b
T2
T4
-DC
1
2
1

C GDP  (  )L
Marco Liserre
T
1
2
 1
C 
 1
L GDP
GDP 

1
C
L
[email protected]
Passivity based control applied to power converters
Passivity Control of the Multilevel Converter

The use of the passivity-based control (PBC) properly fits stability problems related
to this type of converter

Two approaches for the PBC design have been considered

the first is developed considering the overall multilevel converter

the second is developed by splitting the system into n subsystems and
controlling them independently

As regards the second, the partition of the multilevel converter is done on the basis
of energy considerations

The main advantage of the second approach is the separate control of the different
DC-links and a flexible loading capability
Marco Liserre
[email protected]
Passivity based control applied to power converters
Mathematical model of the system

The converter is controlled with a discrete
switching function ui (i=1,2,...,n) for each Hbridge
T11
R
L
T31
+
a
C1
R1
T31
T41
vc1
_
T1i
T3i
+
T3i
T4i
T1n
T3n
ui  T1i  T 3i
i
n

1
x1  (e  Rx1 )   ui x2,i 
L 

i 1
x2,i 
1
 ui x1  i x2,i 
Ci
1 equation
e
Ri
vci
_
n equations
n+1 equations
Marco Liserre
Ci
+
b
T3n
T4n
Cn
Rn
vcn
_
[email protected]
Passivity based control applied to power converters
PBC of an H-bridge
 Since it is possible to control directly the grid current,
z1 
1
(e  Rz1 )   z2 
L
average KVL
on the a.c. side

1
2
 e  Rz  Lz
1
*
1
 RDS z1

predictive action
+
damping injection
 The d.c. voltage is controllable only indirectly, through an internal variable ξ2
z2 
1
  z1   z2 
C
average KCL
on the d.c. side
differential eq.
1
+
ˆ G z
2   z1*  
2
DP 2
C
damping injection

 Then it is necessary to estimate the d.c. load
Marco Liserre




ˆ   ˆ  c2 ˆ  c1  2 z2
[email protected]
Passivity based control applied to power converters
PBC of an n-H-bridge converter

It is not possible a simple extension of the PBC of one H-bridge active
rectifier to n-bridge active rectifier

In fact the PBC of an H-bridge active rectifier needs
one algebraic equation and one differential equation
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC of an n-H-bridge converter

It is not possible a simple extension of the PBC of one H-bridge active
rectifier to n-bridge active rectifier

In fact the PBC of an H-bridge active rectifier needs
one algebraic
equation and one
1 differential
1
ˆ   G z equation
   z  
   e  Rz1  Lz1*  RDS z1 
C

2
*
1
2
DP
2
2

Thus “a simple extension” of this control needs n algebraic equations and n
differential equations. However this is not possible since the n H-bridges
are connected in series on the grid side and the ac voltage equation results
in only one algebraic equation
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC of an n-H-bridge converter

It is not possible a simple extension of the PBC of one H-bridge active
rectifier to n-bridge active rectifier

In fact the PBC of an H-bridge active rectifier needs
1
1
ˆ   G z equation

e  Rz1 equation
Lz1*  RDS z1  and
2 one
  differential
z1*  

one algebraic
2
DP 2
2
C

Thus “a simple extension” of this control needs n algebraic equations and
n differential equations. However this is not possible since the n Hbridges are connected in series on the grid side and the ac voltage
equation results in only one algebraic equation

We have proposed two PBC approaches:
Marco Liserre
1.
one algebraic eq. plus n differential eq. (with ξ2,1= .. = ξ2,n)
2.
n algebraic eq. (based on n virtual KVL’s) plus n differential eq. (with
ξ2,1≠ .. ≠ ξ2,n)
[email protected]
Passivity based control applied to power converters
Passivity-based control approach 1
n
1
e  Rz1  Lz1*  RDS z1
 i 
2
 i 1

1

ˆ G z
 z1*  
 2 
2
DP 2,i for i  1, 2,...., n
C

ˆ   ˆ  c ˆ  c  z
2
1 2 2




Indirect control of output
voltages

i*  I d sin t 



n
Id 

2Vd2  i

2n Vd2
Id 
E
i 1
E




ˆi   i ˆi  c2,i ˆi  c1,i  2,i z2,i


ˆ   ˆ  c2 ˆ  c1  2 z2
1  2  ...  n   & 2,1  2,2  ...  2, n  2
reference voltage
Vd and load power balance reference
current i*
conductance θ
equal for all the bridges
Marco Liserre
internally algebraic switching
function µ
generated ξ2, equal
ODE
for all the bridges
[email protected]
Passivity based control applied to power converters
PBC 2: Model formulation in subsystems
R1
a1
L1
a1
H-Bridge 1
+
Load 1
1 e
i1
i
R1
1 e
L1
H-Bridge 1
+
Load 1
i1
a2
a2
Ri
ai
Li
e
H-Bridge i
+
Load i
i e
ii
ai+1
i1  i2  .... in
Ri
ai
Li
H-Bridge i
+
Load i
i e
n
 i  1
ii
i 1
ai+1
Rn
an
Ln
n e
in
an+1
Marco Liserre
Rn
an
H-Bridge n
+
Load n
n e
Ln
in
H-Bridge n
+
Load n
an+1
[email protected]
Passivity based control applied to power converters
PBC 2: Model formulation in subsystems
R1
a1
L1
R1
a1
H-Bridge 1
+
Load 1
1 e
i1
i
1 e
L1
H-Bridge 1
+
Load 1
i1
a2
a2
Ri
ai
Li
e
H-Bridge i
+
Load i
i e
ii
ai+1
i1  i2  .... in
Ri
ai
Li
H-Bridge i
+
Load i
i e
n
 i  1
ii
i 1
ai+1
Rn
an
Ln
in
an+1
H-Bridge n
+
Load n
Ln
n KVL
 Li x1  i e  Ri x1  ui x2,i
 
Ci x2,i  ui x1  i x2,i
an
n e
Marco Liserre
Rn
ne
in
H-Bridge n
+
Load n
an+1
[email protected]
Passivity based control applied to power converters
Passivity-based control approach 2
Indirect control of each output voltage achieved via the separate
control of each bridge leading to n passivity-based controllers related
through i
i 
Vd2,i i
n 2
 Vd ,i i
i 1
supervisor
Marco Liserre

1



i e  Ri z1  Li z1*  RDS ,i z1
i

 2,i


1
i z1*  ˆi 2,i  GDP ,i z2,i
 2,i 
Ci

ˆ
ˆ
ˆ
i   i i  c2,i i  c1,i  2,i z2,i







for i  1, 2,...., n
for i  1, 2,...., n
for i  1, 2,...., n
n controllers for n H-bridges
[email protected]
Passivity based control applied to power converters
Passivity-based control approach 2
only changing the
parameters of the
controllers
Marco Liserre
[email protected]
Passivity based control applied to power converters
Harmonic compensation
 In case of harmonics, the design results in the use of an RLC active
damping branch very effective in harmonic rejection
 The damping is made by a resistance and a band-pass filter
energy function includes
energy related to harmonics
Marco Liserre
[email protected]
Passivity based control applied to power converters
Harmonic compensation
bandpass
filters
Marco Liserre
[email protected]
Passivity based control applied to power converters
Set-up for the multilevel active rectifier
VLT 5006
GRID
VLT 5006
driving signal
e
& enable
i
vc1
vc2
CONTROLLER D
( space card)
Rated rms grid voltage
Rated power
Reference dc bus voltage
Ac inductance
Dc capacitors
Marco Liserre
220 [V]
1 [kW]
400-560 [V]
10 [mH]
2330 [F]
[email protected]
Passivity based control applied to power converters
PBC tuning: voltage error damping GDP
GDP = 0.1
GDP= 1
310
310
300
300
290
290
280
280
270
270
260
260
250
250
240
240
3
4
5
6
7
8
GDP = 10
3
4
5
6
7
8
dc-link voltage due to a laod step change
Marco Liserre
[email protected]
Passivity based control applied to power converters
PBC tuning: estimate parameter γ
γ = 0.01
γ = 0.01
300
Capacitor voltages [V]
Estimate R1=R2 [Ohm]
350
300
250
200
150
280
260
240
100
50
220
0
2
4
6
Time [s]
0
2
4
6
Time [s]
estimate dc-link load resistance
due to a load step change
it has a strong influence on the dclink dynamic
Marco Liserre
[email protected]
Passivity based control applied to power converters
Steady-state (PBC 1 & 2)
grid voltage
grid current
dc-link voltage
dc-link voltage
Marco Liserre
[email protected]
Passivity based control applied to power converters
Dynamical test
PBC 1
PBC 2
Measured DC voltages [10 V/div] consequent to a load step change
from half to full load on both the DC-links (330 F)
Marco Liserre
[email protected]
Passivity based control applied to power converters
Dynamical test (PBC 2)
dc voltage reference
step on one bus
dc load steps on the two buses
leading to different loads
Measured DC voltages [50 V/div] and grid current [4 A/div] (2330 F)
Marco Liserre
[email protected]
Passivity based control applied to power converters
Dynamical test for active load (PBC 2)
a dc motor has been used
as active load
Marco Liserre
[email protected]
Passivity based control applied to power converters
Unbalance loads on the two dc-links (PBC 2)
Steady-state behavior of PBC2 with full load on DC bus 1 and
half load on DC bus 2
multilevel ac voltage [200 V/div]
Marco Liserre
grid voltage [100V/div] grid
current [10 A/div]
[email protected]
Passivity based control applied to power converters
Computational efforts comparison
15
PBC 2 needs p-3 equations
more than PBC 1
PBC 2
nr. of control equations
12
However PBC 1 employs a
division by the reference
current that leads to
computational problems
9
PBC 1
7
6
5
4
with p = number of desired levels
5
7
9
11
nr. of desired levels
Marco Liserre
[email protected]
Passivity based control applied to power converters
Harmonic compensation
R-damping
RLC-damping
Marco Liserre
[email protected]
Passivity based control applied to power converters
Conclusions

Passivity-based theory offers a straightforward approach to design controllers
without linearazing the system:
 physical and intuitive representation of the control problem
 design method to make the system Lyapunov-stable
 feedback decomposition useful for electromechanical systems

Eulero-Lagrange formulation more suitable for electrical motors

Brayton-Moser formulation more suitable for power converters (tuning procedure)

Optimal results can be obtained with RLC damping of harmonics (similar to those
obtained with generalized integrators – resonant controllers – linear approach)
Marco Liserre
[email protected]